Past

Let $k$ be a field and $A$ a $k$-algebra. The classical Quillen's Lemma states that if $A$ if is equipped with an exhaustive filtration such that the associated graded ring is commutative and finitely generated $k$-algebra then for any finitely generated $A$-module $M$, every element of the endomorphism ring of $M$ is algebraic over $k$. In particular, Quillen's Lemma may be applied to the enveloping algebra of a finite dimensional Lie algebra. I aim to present an affinoid version of Quillen's Lemma which strengthness a theorem proved by Ardakov and Wadsley. Depending on time, I will show how this leads to an (almost) classification of the primitive spectrum of the affinoid enveloping algebra of a semisimple Lie algebra.

Whitney elements on simplices are perhaps the most widely used finite elements in computational electromagnetics. They offer the simplest construction of polynomial discrete differential forms on simplicial complexes. Their associated degrees of freedom (dofs) have a very clear physical meaning and give a recipe for discretizing physical balance laws, e.g., Maxwell’s equations. As interest grew for the use of high order schemes, such as hp-finite element or spectral element methods, higher-order extensions of Whitney forms have become an important computational tool, appreciated for their better convergence and accuracy properties. However, it has remained unclear what kind of cochains such elements should be associated with: Can the corresponding dofs be assigned to precise geometrical elements of the mesh, just as, for instance, a degree of freedom for the space of Whitney 1-forms belongs to a specific edge? We address this localization issue. Why is this an issue? The existing constructions of high order extensions of Whitney elements follow the traditional FEM path of using higher and higher “moments” to define the needed dofs. As a result, such high order finite k-elements in d dimensions include dofs associated to q-simplices, with k < q ≤ d, whose physical interpretation is obscure. The present paper offers an approach based on the so-called “small simplices”, a set of subsimplices obtained by homothetic contractions of the original mesh simplices, centered at mesh nodes (or more generally, when going up in degree, at points of the principal lattice of each original simplex). Degrees of freedom of the high-order Whitney k-forms are then associated with small simplices of dimension k only.  We provide an explicit  basis for these elements on simplices and we justify this approach from a geometric point of view (in the spirit of Hassler Whitney's approach, still successful 30 years after his death).   

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

A dagger category is a category where for every morphism f:x --> y there is a chosen adjoint f*:y --> x, as for example in the category of Hilbert spaces. I will explain this definition in elementary terms and give a few example. The only prerequisites for this talk are the notion of category, functor, and Hilbert space.

Dagger categories are a great categorical framework for some concepts from functional analysis such as C*-algebras and they also allow us to state Atiyah's definition unitary topological field theories in categorical lanugage. There is however a problem with dagger categories: they are what category theorists like to call 'evil'. This is isn't really meant as a moral judgement, it just means that many ways of thinking about ordinary categories don't quite translate to dagger categories.

For example, not every fully faithful and essentially surjective dagger functor is also a dagger equivalence. I will present a notion of 'indefinite completion' that I came up with to describe dagger categories in less 'evil' terms. (Those of you who know Karoubi completion will see a lot of similarities.) I'll also explain how this can be used to compute categories of dagger functors, and more specifically groupoids of unitary TFTs.

Leighton's Theorem states that if two finite graphs have a common universal cover then they have a common finite cover. I will explore various ways in which this result can and can't be extended.

In long-range percolation on $\mathbb{Z}^d$, each potential edge $\{x,y\}$ is included independently at random with probability roughly $\beta\|x-y\|-d-\alpha$, where $\alpha > 0$ controls how long-range the model is and $\beta > 0$ is an intensity parameter. The smaller $\alpha$ is, the easier it is for very long edges to appear. We are normally interested in fixing $\alpha$ and studying the phase transition that occurs as $\beta$ is increased and an infinite cluster emerges. Perhaps surprisingly, the phase transition for long-range percolation is much better understood than that of nearest neighbour percolation, at least when $\alpha$ is small: It is a theorem of Noam Berger that if $\alpha < d$ then the phase transition is continuous, meaning that there are no infinite clusters at the critical value of $\beta$. (Proving the analogous result for nearest neighbour percolation is a notorious open problem!) In my talk I will describe a new, quantitative proof of Berger's theorem that yields power-law upper bounds on the distribution of the cluster of the origin at criticality.
    As a part of this proof, I will describe a new universal inequality stating that on any graph, the maximum size of a percolation cluster is of the same order as its median with high probability. This inequality can also be used to give streamlined new proofs of various classical results on e.g. Erdős-Rényi random graphs, which I will hopefully have time to talk a little bit about also.

I will talk about the joint moments of characteristic polynomials of random unitary matrices and their derivatives. In joint work with Jon Keating and Jon Warren we establish the asymptotics of these quantities for general real values of the exponents as the size N of the matrix goes to infinity. This proves a conjecture of Hughes from 2001. In subsequent joint work with Benjamin Bedert, Mustafa Alper Gunes and Arun Soor we focus on the leading order coefficient in the asymptotics, we connect this to Painleve equations for general values of the exponents and obtain explicit expressions corresponding to the so-called classical solutions of these equations.

Let $G$ be a connected reductive algebraic group, and let $G(\mathbb{F}_q)$ be its group of $\mathbb{F}_q$-rational points. Denote by $\mathrm{Irr}(G(\mathbb{F}_q))$ the set of (equivalence classes) of irreducible finite-dimensional representations. Deligne and Lusztig defined a finite subset $$\mathrm{Unip}(G(\mathbb{F}_q)) \subset \mathrm{Irr}_{\mathrm{fd}}(G(\mathbb{F}_q))$$ 
of unipotent representations. These representations play a distinguished role in the representation theory of $G(\mathbb{F}_q)$. In particular, the classification of $\mathrm{Irr}_{\mathrm{fd}}(G(\mathbb{F}_q))$ reduces to the classification of $\mathrm{Unip}(G(\mathbb{F}_q))$. 

Now replace $\mathbb{F}_q$ with a local field $k$ and replace $\mathrm{Irr}_{\mathrm{fd}}(G(\mathbb{F}_q))$ with $\mathrm{Irr}_{\mathrm{u}}(G(k))$ (irreducible unitary representations). Vogan has predicted the existence of a finite subset 
$$\mathrm{Unip}(G(k)) \subset \mathrm{Irr}_{\mathrm{u}}(G(k))$$ 
which completes the following analogy
$$\mathrm{Unip}(G(k)) \text{ is to } \mathrm{Irr}_{\mathrm{u}}(G(k)) \text{ as } \mathrm{Unip}(G(\mathbb{F}_q)) \text{ is to } \mathrm{Irr}_{\mathrm{fd}}(G(\mathbb{F}_q)).$$
In this talk I will propose a definition of $\mathrm{Unip}(G(k))$ when $k = \mathbb{C}$. The definition is geometric and case-free. The representations considered include all of Arthur's, but also many others. After sketching the definition and cataloging its properties, I will explain a classification of $\mathrm{Unip}(G(\mathbb{C}))$, generalizing the well-known result of Barbasch-Vogan for Arthur's representations. Time permitting, I will discuss some speculations about the case of $k=\mathbb{R}$.

This talk is based on forthcoming joint work with Ivan Loseu and Dmitryo Matvieievskyi.

(joint work with T. Kania, Academy of Sciences of the Czech Republic, Prague)
In this talk we discuss our recent result on the inverse eigenvalue problem for symmetric doubly stochastic matrices. 
Namely, we provide a new sufficient condition for a list of real numbers to be the spectrum of a symmetric doubly stochastic matrix. 
In our construction of such matrices, we employ the eigenvectors of the transition probability matrix of a simple symmetric random walk on the circle. 
We also demonstrate a simple algorithm for generating random doubly stochastic matrices based on our construction. Examples will be provided.

I will present work in progress with D. Gayet and F. Pouran (Grenoble) to establish Russo-Seymour-Welsh (RSW) estimates for 2d statistical mechanics models that do not satisfy the FKG inequality. RSW states that critical percolation has no characteristic length, in the sense that large rectangles are crossed by an open path with a probability that is bounded below by a function of their shape, but uniformly in their size; this ensures the polynomial decay of many relevant quantities and opens the way to deeper understanding of the critical features of the model. All the standard proofs of RSW rely on the FKG inequality, i.e. on the positive correlation between increasing events; we establish the stability of RSW under small perturbations that do not preserve FKG, which extends it for instance to the high-temperature anti-ferromagnetic Ising model.

Subfactors and K-theory are useful mechanisms for understanding modular tensor categories and conformal field theories. As part of this programme, one issue to try and construct or reconstruct a conformal field theory as the representation theory of a conformal net of algebras, or as a vertex operator algebra from a given abstractly presented modular tensor category. Orbifold models play an important role and orbifolds of Tambara-Yamagami systems are relevant to understanding the double of the Haagerup as a conformal field theory. This is joint work with Andreas Aaserud, Terry Gannon and Ulrich Pennig.

I will review recent works on geometries underlying scattering amplitudes of (certain generalizations of) particles and strings  Tree amplitudes of a cubic scalar theory are given by "canonical forms" of the so-called ABHY associahedra defined in kinematic space. The latter can be naturally extended to generalized associahedra for finite-type cluster algebra, and for classical types their canonical forms give scalar amplitudes through one-loop order. We then consider vast generalizations of string amplitudes dubbed “stringy canonical forms”, and in particular "cluster string integrals" for any Dynkin diagram, which for type A reduces to usual string amplitudes. These integrals enjoy remarkable factorization properties at finite $\alpha'$, obtained simply by removing nodes of the Dynkin diagram; as $\alpha'\rightarrow 0$ they reduce to canonical forms of generalized associahedra, or the aforementioned tree and one-loop scalar amplitudes.

It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations (PDEs) and most of the numerical approximation methods for PDEs in the scientific literature suffer from the so-called curse of dimensionality (CoD) in the sense that the number of computational operations employed in the corresponding approximation scheme to obtain an  approximation precision $\varepsilon >0$ grows exponentially in the PDE dimension and/or the reciprocal of $\varepsilon$. Recently, certain deep learning based approximation methods for PDEs have been proposed  and various numerical simulations for such methods suggest that deep neural network (DNN) approximations might have the capacity to indeed overcome the CoD in the sense that  the number of real parameters used to describe the approximating DNNs  grows at most polynomially in both the PDE dimension $d \in  \N$ and the reciprocal of the prescribed approximation accuracy $\varepsilon >0$. There are now also a few rigorous mathematical results in the scientific literature which  substantiate this conjecture by proving that  DNNs overcome the CoD in approximating solutions of PDEs.  Each of these results establishes that DNNs overcome the CoD in approximating suitable PDE solutions  at a fixed time point $T >0$ and on a compact cube $[a, b]^d$ but none of these results provides an answer to the question whether the entire PDE solution on $[0, T] \times [a, b]^d$ can be approximated by DNNs without the CoD. 
In this talk we show that for every $a \in \R$, $ b \in (a, \infty)$ solutions of  suitable  Kolmogorov PDEs can be approximated by DNNs on the space-time region $[0, T] \times [a, b]^d$ without the CoD. 

We discuss the problem in representation theory of decomposing restricted representations. We start classically with the symmetric groups via Young diagrams and Young tableaux, and then move into the world of Lie groups. These problems have connections with both physics and number theory, and if there is time I will discuss the Gan-Gross-Prasad conjectures which predict results on restrictions for algebraic groups over both local and global fields. The pre-requisites will build throughout the talk, but it should be accessible to anyone with some knowedge of both finite groups and Lie groups.

Hardt-Simon proved that every area-minimizing hypercone $C$ having only an isolated singularity fits into a foliation of $R^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $C$. We prove that if a minimal hypersurface $M$ in the unit ball $B_1 \subset R^{n+1}$ lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone), then $M \cap B_{1/2}$ is a $C^{1,\alpha}$ perturbation of either the cone itself, or some leaf of its associated foliation. In particular, we show that singularities modeled on these cones determine the local structure not only of $M$, but of any nearby minimal surface. Our result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces in $R^{n+1}$ asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation.  This is joint work with Luca Spolaor.

Triangle presentations are combinatorial structures on finite projective geometries which characterize groups acting simply transitively on the vertices of locally finite affine A_n buildings. From this data, we will show how to construct new fiber functors on the category of tilting modules for SL(n+1) in characteristic p (related to order of the projective geometry) using the web calculus of Cautis, Kamnitzer, Morrison and Brundan, Entova-Aizenbud, Etingof, Ostrik.

In this talk, I will discuss some results on the structure of the cohomology of the moduli space of stable SL_n Higgs bundles on a curve. 

One consequence is a new proof of the Hausel-Thaddeus conjecture proven previously by Groechenig-Wyss-Ziegler via p-adic integration.

We will also discuss connections to the P=W conjecture if time permits. Based on joint work with Junliang Shen.

In view of the recent observations of gravitational-wave signals from black-hole mergers, classical black-hole scattering has received considerable interest due to its relation to the classical bound-state problem of two black holes inspiraling onto each other. In this talk I will discuss the link between classical scattering of spinning black holes and quantum scattering amplitudes for massive spin-s particles. Considering the first post-Minkowskian (PM) order, I will explain how the spin-exponentiated structure of the relevant tree-level amplitude follows from minimal coupling to Einstein's gravity and in the s → ∞ limit generates the black holes' complete series of spin-induced multipoles. The resulting scattering function will be shown to encode in a simple way the classical net changes in the black-hole momenta and spins at 1PM order and to all orders in spins. I will then comment on the results and challenges at 2PM order and beyond.
 

We propose Swampland constraints on consistent 5d N=1 supergravity theories. In particular, we focus on a special class of BPS monopole strings which arise only in gravitational theories. The central charges and the levels of current algebras of 2d CFTs on these strings can be computed using the anomaly inflow mechanism and provide constraints for the 5d supergravity using unitarity of the worldsheet CFT. In M-theory, where these theories can be realised by compactification on Calabi-Yau threefolds, the special monopole strings arise from M5 branes wrapping “semi-ample” 4-cycles in the threefolds. We further identify necessary geometric conditions that such cycles need to satisfy and translate them into constraints for the low-energy gravity theory.

In this talk, I will give a brief introduction to level-set methods for image analysis. I will then describe an application of level-sets to the construction of filtrations for persistent homology computations. I will present several case studies with various spatial data sets using this construction, including applications to voting, analyzing urban street patterns, and spiderwebs. I will conclude by discussing the types of data which I might imagine such methods to be suitable for analyzing and suggesting a few potential future applications of level-set based computations.

The Schur functor and its inverses give an important connection between the representation theories of the symmetric group and the general linear group. Kleshchev and Nakano proved in 2001 that when the characteristic of the field is at least 5, the image of the Specht module under the inverse Schur functor is isomorphic to the dual Weyl module. In this talk I will address what happens in characteristics 2 and 3: in characteristic 3, the isomorphism holds, and I will give an elementary proof of this fact which covers also all characteristics other than 2; in characteristic 2, the isomorphism does not hold for all Specht modules, and I will classify those for which it does. Our approach is with Young tableaux, tabloids and Garnir relations.

In this talk, I will present a series of new experimental data, supported by theoretical models, of the transport of ash, aerosols and bubbles in multiphase plumes rising through stratified environments, focussing on the structure of flow and the dispersal of the different phases. The models have relevance for the dispersal of volcanic ash in the atmosphere and ocean, the mixing of aerosols in buildings, and the fate of suspended sediment produced during deep sea mining. 

While the pathological mechanisms in COVID-19 illness are still poorly understood, it is increasingly clear that high levels of pro-inflammatory mediators play a major role in clinical deterioration in patients with severe disease. Current evidence points to a hyperinflammatory state as the driver of respiratory compromise in severe COVID-19 disease, with a clinical trajectory resembling acute respiratory distress syndrome (ARDS) but how this “runaway train” inflammatory response emergences and is maintained is not known. In this talk, we present the first mathematical model of lung hyperinflammation due to SARS- CoV-2 infection. This model is based on a network of purported mechanistic and physiological pathways linking together five distinct biochemical species involved in the inflammatory response. Simulations of our model give rise to distinct qualitative classes of COVID-19 patients: (i) individuals who naturally clear the virus, (ii) asymptomatic carriers and (iii–v) individuals who develop a case of mild, moderate, or severe illness. These findings, supported by a comprehensive sensitivity analysis, points to potential therapeutic interventions to prevent the emergence of hyperinflammation. Specifically, we suggest that early intervention with a locally-acting anti-inflammatory agent (such as inhaled corticosteroids) may effectively blockade the pathological hyperinflammatory reaction as it emerges.

Generative adversarial networks (GANs) have enjoyed tremendous success in image generation and processing, and have recently attracted growing interests in other fields of applications. In this talk we will start from analyzing the connection between GANs and mean field games (MFGs) as well as optimal transport (OT). We will first show a conceptual connection between GANs and MFGs: MFGs have the structure of GANs, and GANs are MFGs under the Pareto Optimality criterion. Interpreting MFGs as GANs, on one hand, will enable a GANs-based algorithm (MFGANs) to solve MFGs: one neural network (NN) for the backward Hamilton-Jacobi-Bellman (HJB) equation and one NN for the Fokker-Planck (FP) equation, with the two NNs trained in an adversarial way. Viewing GANs as MFGs, on the other hand, will reveal a new and probabilistic aspect of GANs. This new perspective, moreover, will lead to an analytical connection between GANs and Optimal Transport (OT) problems, and sufficient conditions for the minimax games of GANs to be reformulated in the framework of OT. Building up from the probabilistic views of GANs, we will then establish the approximation of GANs training via stochastic differential equations and demonstrate the convergence of GANs training via invariant measures of SDEs under proper conditions. This stochastic analysis for GANs training can serve as an analytical tool to study its evolution and stability.

I will give an introduction to semigroup C*-algebras of ax+b-semigroups over rings of algebraic integers in algebraic number fields, a class of C*-algebras that was introduced by Cuntz, Deninger, and Laca. After explaining the construction, I will briefly discuss the state-of-the-art for this example class: These C*-algebras are unital, separable, nuclear, strongly purely infinite, and have computable primitive ideal spaces. In many cases, e.g., for Galois extensions, they completely characterise the underlying algebraic number field.

Polycyclic groups form an interesting and well-studied class of groups that properly contain the finitely generated nilpotent groups. I will discuss the C*-algebras associated with virtually polycyclic groups, their maximal quotients and recent work with Jianchao Wu showing that they have finite nuclear dimension.

Artificial microswimmers are an emerging field of research, attracting
interest as testing beds for physical theories of complex biological
entities, as inspiration for the design of smart materials, and for the
sheer elegance, and often quite counterintuitive phenomena of
experimental nonlinear dynamics.

Self-propelling droplets are among the most simplified swimmer models
imaginable, requiring just three components (oil, water, surfactant). In
this talk, I will show how these inherently stupid objects can make
surprisingly smart decisions based on interactions with microfluidic
structures and self-generated and external chemical fields.

I will present a simple model of market microstructure which explains the concavity of price impact. In the proposed model, the local relationship between the order flow and the fundamental price (i.e. the local price impact) is linear, with a constant slope, which makes the model dynamically consistent. Nevertheless, the expected impact on midprice from a large sequence of co-directional trades is nonlinear and asymptotically concave. The main practical conclusion of the model is that, throughout a meta-order, the volumes at the best bid and ask prices change (on average) in favor of the executor. This conclusion, in turn, relies on two more concrete predictions of the model, one of which can be tested using publicly available market data and does not require the (difficult to obtain) information about meta-orders. I will present the theoretical results and will support them with the empirical analysis.

We briefly review mathematical models of viscous deformable interfaces (such as plasma membranes) leading to fluid equations posed on (evolving) 2D surfaces embedded in $R^3$. We further report on some recent advances in understanding and numerical simulation of the resulting fluid systems using an unfitted finite element method.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

Solving kinetic or related models with high-dimensional random parameters has been a challenging problem. In this talk, we will discuss how to employ the bi-fidelity stochastic collocation and choose efficient low-fidelity models in order to solve a class of multi-scale kinetic equations with uncertainties, including the Boltzmann equation, linear transport and the Vlasov-Poisson equation. In addition, some error analysis for the bi-fidelity method based on these PDEs will be presented. Finally, several numerical examples are shown to validate the efficiency and accuracy of the proposed method.

Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size ω1 but with no uncountable branches play an important role by being clocks of Ehrenfeucht–Fraïssé games that measure similarity of model of size ℵ1. We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that map T into T’ while preserving the strict order <_T. This order translates into the comparison of winning strategies for the isomorphism player, where any winning strategy for T’ translates into a winning strategy for T’. Hence it is natural to ask if there is a largest such tree, or as we would say, a universal tree for the class of wood Aronszajn trees with weak embeddings. It was known that there is no such a tree under CH, but in 1994 Mekler and Väänanen conjectured that there would be under MA(ω1).

In our upcoming JSL  paper with Saharon Shelah we prove that this is not the case: under MA(ω1) there is no universal wide Aronszajn tree.

The talk will discuss that paper. The paper is available on the arxiv and on line at JSL in the preproof version doi: 10.1017/jsl.2020.42.

I will discuss the notion of invariably generated groups, its importance, and some intuition. I will then present a construction of an invariably generated group that admits an index two subgroup that is not invariably generated. The construction answers questions of Wiegold and of Kantor-Lubotzky-Shalev. This is a joint work with Nir Lazarovich.

In this talk, I will study the properties of parametric estimators based on the Maximum Mean Discrepancy (MMD) defined by Briol et al. (2019). In a first time, I will show that these estimators are universal in the i.i.d setting: even in case of misspecification, they converge to the best approximation of the distribution of the data in the model, without ANY assumption on this model. This leads to very strong robustness properties. In a second time, I will show that these results remain valid when the data is not independent, but satisfy instead a weak-dependence condition. This condition is based on a new dependence coefficient, which is itself defined thanks to the MMD. I will show through examples that this new notion of dependence is actually quite general. This talk is based on published works, and works in progress, with Badr-Eddine Chérief Abdellatif (ENSAE Paris), Mathieu Gerber (University of Bristol), Jean-David Fermanian (ENSAE Paris) and Alexis Derumigny (University of Twente):

http://arxiv.org/abs/1912.05737

http://proceedings.mlr.press/v118/cherief-abdellatif20a.html

http://arxiv.org/abs/2006.00840

https://arxiv.org/abs/2010.00408

https://cran.r-project.org/web/packages/MMDCopula/

A connected matching is a matching contained in a connected component. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs.
    
I will describe Łuczak's reduction, introduce the new reduction, and mention potential applications of the improved method.

This talk is intended for a broad math and physics audience in particular including students. It will focus on the speaker’s recent contributions to the analysis of the real Ginibre ensemble consisting of square real matrices whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre ensemble attain positive likelihood. In turn, the spectral radius of a real Ginibre matrix follows a different limiting law for purely real eigenvalues than for non-real ones. We will show that the limiting distribution of the largest real eigenvalue admits a closed form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov-Shabat system. This system is directly related to several of the most interesting nonlinear evolution equations in 1 + 1 dimensions which are solvable by the inverse scattering method. The results of this talk are based on our joint work with Jinho Baik (arXiv:1808.02419 and arXiv:2008.01694)

In this talk I will discuss relations between the low-energy expansions  of open- and closed string amplitudes. At genus zero, it has been shown that the single-valued map of MZVs maps open-string amplitudes to their closed-string counterparts. After reviewing this story, I will discuss recent work at genus one which aims to define a similar mapping from the open to the closed string. Our construction is driven by the differential equations and degeneration limits of certain generating functions of string integrals and suggests a pairing of integration cycles and forms at genus one - analogous to the duality between Parke-Taylor factors and disk boundaries at genus zero. Finally, I will discuss the impact of said mapping on the elliptic MZVs and modular graph forms which arise naturally upon solving these differential equations.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

In 2013, Reeder–Yu gave a construction of supercuspidal representations by starting with stable characters coming from the shallowest depth of the Moy–Prasad filtration. In this talk, we will be diving deeper—but not too deep. In doing so, we will construct examples of supercuspidal representations coming from a larger class of “shallow” characters. Using methods similar to Reeder–Yu, we can begin to make predictions about the Langlands parameters for these representations.

Here we propose a new method to compare the modular structure of a pair of node-aligned networks. The majority of current methods, such as normalized mutual information, compare two node partitions derived from a community detection algorithm yet ignore the respective underlying network topologies. Addressing this gap, our method deploys a community detection quality function to assess the fit of each node partition with respect to the other network's connectivity structure. Specifically, for two networks A and B, we project the node partition of B onto the connectivity structure of A. By evaluating the fit of B's partition relative to A's own partition on network A (using a standard quality function), we quantify how well network A describes the modular structure of B. Repeating this in the other direction, we obtain a two-dimensional distance measure, the bi-directional (BiDir) distance. The advantages of our methodology are three-fold. First, it is adaptable to a wide class of community detection algorithms that seek to optimize an objective function. Second, it takes into account the network structure, specifically the strength of the connections within and between communities, and can thus capture differences between networks with similar partitions but where one of them might have a more defined or robust community structure. Third, it can also identify cases in which dissimilar optimal partitions hide the fact that the underlying community structure of both networks is relatively similar. We illustrate our method for a variety of community detection algorithms, including multi-resolution approaches, and a range of both simulated and real world networks.

Let $X$ be a random variable, taking values in $\{1,…,n\}$, with standard deviation $\sigma$ and let $f_X$ be its probability generating function. Pemantle conjectured that if $\sigma$ is large and $f_X$ has no roots close to 1 in the complex plane then $X$ must approximate a normal distribution. In this talk, I will discuss a complete resolution of Pemantle's conjecture. As an application, we resolve a conjecture of Ghosh, Liggett and Pemantle by proving a multivariate central limit theorem for, so called, strong Rayleigh distributions. I will also discuss how these sorts of results shed light on random variables that arise naturally in combinatorial settings. This talk is based on joint work with Marcus Michelen.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

Michael Coughlan (with Sam Howison, Ian Hewitt, Andrew Wells)

Arctic sea ice forms a thin but significant layer at the ocean surface, mediating key climate feedbacks. During summer, surface melting produces considerable volumes of water, which collect on the ice surface in ponds. These ponds have long been suggested as a contributing factor to the discrepancy between observed and predicted sea ice extent. When viewed at large scales ponds have a complicated, approximately fractal geometry and vary in area from tens to thousands of square meters. Increases in pond depth and area lead to further increases in heat absorption and overall melting, contributing to the ice-albedo feedback. 

Previous modelling work has focussed either on the physics of individual ponds or on the statistical behaviour of systems of ponds. In this talk I present a physically-based network model for systems of ponds which accounts for both the individual and collective behaviour of ponds. Each pond initially occupies a distinct catchment basin and evolves according to a mass-conserving differential equation representing the melting dynamics for bare and water-covered ice. Ponds can later connect together to form a network with fluxes of water between catchment areas, constrained by the ice topography and pond water levels. 

I use the model to explore how the evolution of pond area and hence melting depends on the governing parameters, and to explore how the connections between ponds develop over the melt season. Comparisons with observations are made to demonstrate the ways in which the model qualitatively replicates properties of pond systems, including fractal dimension of pond areas and two distinct regimes of pond complexity that are observed during their development cycle. 

Different perimeter-area relationships exist for ponds in the two regimes. The model replicates these relationships and exhibits a percolation transition around the transition between these regimes, a facet of pond behaviour suggested by previous studies. The results reinforce the findings of these studies on percolation thresholds in pond systems and further allow us to constrain pond coverage at this threshold - an important quantity in measuring the scale and effects of the ice-albedo feedback.

After a review of Batalin-Vilkovisky formalism and homotopy algebras, we discuss how these structures emerge in quantum field theory and gravity. We focus then on the application of these sophisticated mathematical tools to scattering amplitudes (both tree- and loop-level) and to the understanding of the dualities between gauge theories and gravity, highlighting generalizations of old results and presenting new ones.

In this talk I will give an introduction to random multiplicative functions, and cover the recent developments in this area. I will also explain how RMF's are connected to some of the important open problems in Analytic Number Theory.

The Ricci flow on a surface is an intrinsic evolution of the metric converging to a constant curvature metric within the conformal class. It can be seen as an infinite-dimensional gradient flow. We introduce a natural 'Langevin' version of that flow, thus constructing an SPDE with invariant measure expressed in terms of Liouville Conformal Field Theory.
Joint work with Hao Shen (Wisconsin).

The main result is the existence of smooth, properly embedded 3-discs in S¹ × D³ that are not smoothly isotopic to {1} × D³. We describe a 2-variable Laurent polynomial invariant of 3-discs in S¹ × D³. This allows us to show that, when taken up to isotopy, such 3-discs form an abelian group of infinite rank. Joint work with David Gabai.

We describe how Smith theory applies in the setting of Hamiltonian Floer homology filtered by the action functional, and provide applications to questions regarding Hamiltonian diffeomorphisms, including the Hofer-Zehnder conjecture on the existence of infinitely many periodic points and a question of McDuff-Salamon on Hamiltonian diffeomorphisms of finite order.

All five-dimensional non-abelian gauge theories have a U(1)U(1)I​U(1) global symmetry associated with instantonic particles. I will describe a mixed ’t Hooft anomaly between this and other global symmetries of  the theory, namely the one-form center symmetry or ordinary flavor symmetry for theories with fundamental matter. I will explore some general dynamical properties of the candidate phases implied by the anomaly, and apply our results to supersymmetric gauge theories in five dimensions, analysing the symmetry enhancement patterns occurring at their conjectured RG fixed points.